Abstract
Consider a simple symmetric random walk on the integer lattice $\ZB$. For each n, let $V(n)$ denote a favorite site (or most visited site) of the random walk in the first n steps. A somewhat surprising theorem of Bass and Griffin [Z. Wahrsch. Verw. Gebiete 70 (1985) 417--436] says that V is almost surely transient, thus disproving a previous conjecture of Erdős and Révész [Mathematical Structures--Computational Mathematics--Mathematical Modeling 2 (1984) 152--157]. More precisely, Bass and Griffin proved that almost surely, $\liminf_{n\to \infty} {|V(n)| \over n^{1/2}(\log n)^{-\gamma}}$ equals $0$ if $\gamma<:1$, and is infinity if $\gamma>11$ (eleven). The present paper studies the rate of escape of $V(n)$. We show that almost surely, the "lim\,inf'' expression in question is 0 if $\gamma\leq 1$, and is infinity otherwise. The corresponding problem for Brownian motion is also studied.
Citation
Mikhail A. Lifshits. Zhan Shi. "The escape rate of favorite sites of simple random walk and Brownian motion." Ann. Probab. 32 (1A) 129 - 152, January 2004. https://doi.org/10.1214/aop/1078415831
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